n this article we investigate numerically the spectrum of some representative
examples of discrete one-dimensional Schrödinger operators with quasi-periodic potential
in terms of a perturbative constant b and the spectral parameter a. Our examples
include the well-known Almost Mathieu model, other trigonometric potentials with a single
quasi-periodic frequency and generalisations with two and three frequencies. We computed
numerically the rotation number and the Lyapunov exponent to detect open and collapsed
gaps, resonance tongues and the measure of the spectrum. We found that the case with one
frequency was significantly different from the case of several frequencies because the latter
has all gaps collapsed for a sufficiently large value of the perturbative constant and thus the
spectrum is a single spectral band with positive Lyapunov exponent. In contrast, in the cases
with one frequency considered, gaps are always dense in the spectrum, although some gaps
may collapse either for a single value of the perturbative constant or for a range of values. In
all cases we found that there is a curve in the (a, b)-plane which separates the regions where
the Lyapunov exponent is zero in the spectrum and where it is positive. Along this curve,
which is b = 2 in the Almost Mathieu case, the measure of the spectrum is zero.